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Re: Norm

  • To: mathgroup at smc.vnet.net
  • Subject: [mg68005] Re: Norm
  • From: Bill Rowe <readnewsciv at earthlink.net>
  • Date: Thu, 20 Jul 2006 06:04:46 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

On 7/19/06 at 5:21 AM, Clausenator at gmail.com wrote:

>Hi, I want to calculate a distance matrix, similar to (as poorly
>explained at) http://en.wikipedia.org/wiki/Distance_matrix

>I found out about the Function "Norm" in mathematica 5.

>Here is a little example. I want to calculate the distance between
>vectors {0,1} and {5,1}. The distance should be 5

>Now,

>Norm[{{0., 1.}, {5., 1.}}, 2] results 5.10293

>Norm[{{0., 1.} - {5., 1.}}, 2] results 5.0

>According to the documentation I have (Mathematica Help Browser,
>search for "Norm" under "Built-in Functions") the version with the
>comma is documented. I like the solution with the dash better. Which
>one is it? In other words, is there some Wolfram description or can
>you explain the difference?

Yes,

In[10]:=
{{0.,1.},{5.,1.}}!={{0.,1.}-{5.,1.}}

Out[10]=
True

Norm[{{0,.1,}-{5.,1.}},2] is exactly the same as Norm[{{5.,0}},2] which is 5. The dash tells Mathematica to do a subtraction then compute the norm.

Norm[{{0.,1.},{5.,1.}},2] is the norm of a 2x2 matrix and is not equal to 5. In particular for a matrix, m, Norm[m] is a singular value of m. Also, the default for the Norm function is the 2-norm. That is

In[4]:=
Norm[{{0,1},{5,1}}]==Norm[{{0,1},{5,1}},2]

Out[4]=
True

and

In[9]:=
Norm[{{0,1},{5,1}}//N]==First@SingularValueList[{{0,1},{5,1}}//N]

Out[9]=
True

Finally, all of this is documented and can be found using the Help Browser.
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